Atoms 2015, 3, 2-52; doi:10.3390/atoms3010002
OPEN ACCESS
atoms ISSN 2218-2004 www.mdpi.com/journal/atoms Article
Ab-Initio Calculations of Level Energies, Oscillator Strengths and Radiative Rates for E1 Transitions in Beryllium-Like Iron Ahmed Abou El-Maaref 1,2, *, Stefan Schippers 2 and Alfred Müller 2 1 2
Physics Department, Faculty of Science, Al-Azhar University Assiut, 71524 Assiut, Egypt Institut für Atom- und Molekülphysik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany; E-Mails:
[email protected] (S.S.);
[email protected] (A.M.)
* Author to whom correspondence should be addressed; E-Mail:
[email protected]; Tel.: +20-1018918939. Academic Editor: James F. Babb Received: 10 November 2014 / Accepted: 9 January 2015 / Published: 20 January 2015
Abstract: In the present work, energy levels, oscillator strengths, radiative rates and wavelengths of Be-like iron (Fe22+ ) from ab-initio calculations using the multiconfiguration Dirac-Hartree-Fock method are presented. These quantities have been calculated for a set of configurations in the general form 1s2 nl n0 l0 , where n = 2, 3 and n0 = 2, 3, 4, 5 and l = s, p, d and l0 = s, p, d, f, g. In addition, excitations of up to four electrons, including core-electron excitations, have been considered to improve the quality of the wave functions. This study comprises an extensive set of E1 transition rates between states with different J. The present results are compared with the available experimental and theoretical data. Keywords: energy levels; oscillator strengths; radiative rates; be-like iron
1. Introduction Accurate atomic data for iron ions are of interest in astrophysics, especially for the identification of solar spectra [1–4], as well as in the physics of controlled fusion [5] and plasma diagnostic [6]. From the astrophysical point of view, the importance of iron ions lies in the fact that iron is the cosmically most abundant heavy element beyond silicon [7]. The beryllium isoelectronic sequence including the Fe22+ ion has been studied using different theoretical approaches [8–21]. Most of the earlier calculations have
Atoms 2015, 3
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produced results for one- or two-electron excitations to low-lying levels (up to n = 2 or 3). Excitation to high-lying levels was studied by Moribayashi and Kato [22], including configurations up to 2pnl (n ≤ 20, l = s, p, d) using Cowan’s code [23]. Recently, many theoretical calculations have been carried out for Fe22+ to meet the needs for accurate atomic structure data. Chidichimo et al. have calculated level energies for n ≤ 4, as well as wavelengths and weighted oscillator strengths using the Belfast R-matrix programs [24,25]. Del Zanna et al. [26] have compiled experimental observations of energy levels and wavelengths, and performed calculations of weighted oscillator strengths for Be-like iron using the non-relativistic SUPERSTRUCTURE program. Santos et al. [27] calculated probabilities for transitions from the 1s2 2s3p 3 P0 level for selected beryllium-like ions, from Z = 5 to 92. They used the MCDF method including relativistic effects, QED (quantum-electrodynamics) effects, and correlations up to the 4f subshell, however, they neglected the Breit interaction. Jian-Hui et al. [28] have calculated energy levels, oscillator strengths, transition probabilities and wavelengths of Fe22+ using the MCDF method with the inclusion of vacuum polarization and Breit interaction. Charro et al. [29,30] have calculated some oscillator strengths of Be-like iron up to n = 2, 3 using the relativistic quantum defect orbital (RQDO) method and the MCDF method, but there were no Breit corrections included in the calculations. By using the FAC code Landi and Gu [31] calculated energy levels, oscillator strengths and transition probabilities 0 for 166 fine-structure levels of Fe22+ belonging to the complexes 1s2 2lnl , n = 2 − 5, l = s, p, d, f, g. In the present work an extensive set of configuration state functions (CSF) including subshell populations up to the 5g subshell is used. In addition, the Breit interaction and QED effects are incorporated, which have been neglected in most of the previous calculations. The present comprehensive treatment of the Fe22+ atomic structure aims at providing more accurate results than hitherto available. For the present extensive atomic structure calculations of beryllium-like iron, Fe22+ , we have used the multiconfiguration Dirac-Fock (MCDF) method [32] as implemented in the GRASP2K code [33]. Excitations from n = 3 to n = 4, 5 (doubly-excited levels) are included, and EOL (extended optimal level) type calculations have been performed. Wavelengths, energy levels, and E1 transition parameters (oscillator strengths, transition probabilities, and line strengths) have been computed for 182 fine-structure levels. The calculations have been divided into two main groups, with even and odd parity. The odd-parity group contains 90 levels while the even-parity group has 92 levels. The present calculations of oscillator strengths and radiative rates are generally in a good agreement with corresponding values in the NIST atomic data compilation [34]. The good agreement between our length and velocity gauge values provides some indication (although not a sufficient one) for the accuracy of the wave functions used in the present study. 2. Method of Calculation Details of the MCDF method as implemented in the GRASP2K code can be found in References [32,33]. For the nuclear charge distribution within the 56 Fe nucleus, we used the default Fermi distribution parameters suggested in GRASP2K. The initial estimate for the radial orbitals is generated by solving the Dirac equation in a Thomas-Fermi potential for a single reference configuration (i.e., the 2s2 level for even levels and the 2s2p level for odd levels) by allowing the single, double,
Atoms 2015, 3
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triple, and quadruple excitations to active orbital sets with n = 2, 3, 4, 5. The self-consistent procedure (RSCF) including EOL type calculations (extended optimal levels) is done layer by layer, at each stage the outer orbitals are optimized. The EOL type calculations construct orbitals from an average energy functional in which the fine-structure levels are given the weight (2J + 1) [30]. This procedure is performed for every J-value separately. The splitting of the atomic levels into different groups has been found to be a useful compromise between two basic requirements in the atomic calculations, first, to get accurate wave functions for the radiative-rate calculations, and second, to keep the procedures manageable even with a large number of CSFs [35]. The same computational method has been applied to the even parity levels. The RSCF calculations were followed by relativistic configuration interaction (RCI) calculations including the Breit interaction Hamiltonian [36]. For the Breit interaction, we used the default low-frequency-limit approximation of the first-order perturbation theory, as implemented in the GRASP2K code. The GRASP2K procedure JJ2LSJ was used for the transformation of ASFs (atomic state functions) from a jj-coupled CSF basis into a LSJ-coupled CSF basis [37]. The beryllium-like iron atomic system has four electrons. In the ground state two electrons reside in the closed K-shell and the other two in the closed 2s subshell. In the first step of building the ground state wave function of the beryllium-like ion, only interactions between the two outer shell electrons were considered. In a second step the interaction with the atomic core was additionally considered since it is very important for the calculation of the wave functions of the excited states . The ground level of the Fe22+ ion is the 1s2 2s2 1 S0 level, and the excited levels under consideration in this work belong to 1s2 nl n0 l0 configurations, where n = 2, 3 and n0 = 2, 3, 4, 5 and l = s, p, d and l0 = s, p, d, f, g with different angular momenta and parities. The open K-shell states are already included in our calculations as admixing correlations, but not explicitly provided. We categorized these levels into groups having the same angular momentum and parity. For example the even parity states with J = 3 are represented by 57161 jj-coupled CSF. As shown in Table 1, the wave function expansions increase rapidly in size by increasing nl which means that we can get unpractically high numbers of CSFs for n > 5. The numbers of CSFs which are generated by quadruple excitations are shown in Table 1 which illustrates the degree of complexity of the present calculations. Table 1. Number of configuration state functions (CSFs) used in the atomic state function expansion for the given angular momentum and parity (J P ) considering only quadruple excitations. J+
3l
4l
5l
J−
3l
4l
5l
0 1 2 3 4 5
211 436 534 380 228 89
2149 5384 7250 6930 5588 3650
13,592 36,634 52,481 57,161 53,512 43,358
0 1 2 3 4 5
180 460 516 392 222 90
2040 5476 7168 6988 5540 3672
13,302 36,894 52,238 57,354 53,342 43,466
3. Results and Discussion The calculated total energies (in a.u.) and energy levels (in eV) are shown in ascending order in Table 2 where also comparisons with literature values [26,34,38,39] are included. Our calculated level
Atoms 2015, 3
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energies for Fe22+ are in good agreement with the NIST levels [34]. The relative deviation is generally better than 1.0%, except for the levels (2); 1s2 2s2p(3 P0 ) and (3); 1s2 2s2p(3 P1 ), where the relative deviations from NIST energies are 1.16% and 1.21%, respectively. These deviations may be due to the limited numbers of CSFs in the calculations. However, including more correlations by adding 6l orbitals to the expansion would produce a number of CSFs greater than our computer memory can tolerate. As an illustration, the number of CSFs generated by quadruple (Q) excitation to 6l orbitals with J P = 2+ is 260,702. The energy of level 33, 1s2 2p3d(1 D2 ), differs from the corresponding NIST [34] value by 0.13%, while the values by Del Zanna et al. [26] and the CHIANTI database [38] deviate from the NIST [34] value by 0.94% and 0.97%, respectively. Table 2. Total energies Etotal (in a.u.) and energy levels (in eV) of Be-like iron (Fe22+ ). The sixth column lists the present energy levels, and the next 5-columns provide the energies from NIST database [34], the observed energies by Del Zanna et al. [26] and Gu et al. [39], and the calculated (Theor.) and observed (Exp.) values from the CHIANTI database [38]. Key
Configuration
J
Parity
Etotal
Present
NIST
Reference [26]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1s2 2s2 (1 S) 1s2 2s2p(3 P ) 1s2 2s2p(3 P ) 1s2 2s2p(3 P ) 1s2 2s2p(1 P ) 1s2 2p2 (3 P ) 1s2 2p2 (3 P ) 1s2 2p2 (3 P ) 1s2 2p2 (1 D) 1s2 2p2 (1 S) 1s2 2s3s(3 S) 1s2 2s3s(1 S) 1s2 2s3p(3 P ) 1s2 2s3p(3 P ) 1s2 2s3p(1 P ) 1s2 2s3p(3 P ) 1s2 2s3d(3 D) 1s2 2s3d(3 D) 1s2 2s3d(3 D) 1s2 2s3d(1 D) 1s2 2p3s(3 P ) 1s2 2p3s(3 P ) 1s2 2p3p(3 D) 1s2 2p3s(3 P ) 1s2 2p3s(1 P ) 1s2 2p3p(3 D) 1s2 2p3p(1 P ) 1s2 2p3p(3 P ) 1s2 2p3d(3 F ) 1s2 2p3p(3 P ) 1s2 2p3d(3 F ) 1s2 2p3p(3 D) 1s2 2p3d(1 D) 1s2 2p3p(3 S) 1s2 2p3p(3 P )
0 0 1 2 1 0 1 2 2 0 1 0 1 0 1 2 1 2 3 2 0 1 1 2 1 2 1 0 2 1 3 3 2 1 2
+ – – – – + + + + + + + – – – – + + + + – – + – – + + + – + – + – + +
–812.1123386 –810.5074865 –810.4056986 –809.9784327 –808.6849085 –807.7638371 –807.4272328 –807.2353664 –806.6250012 –805.6167768 –771.5817273 –771.2425609 –770.8241423 –770.779389 –770.679708 –770.6540603 –770.2600664 –770.2535233 –770.1730742 –769.9289828 –769.5356786 –769.495422 –769.0671809 –769.0006032 –768.7864995 –768.7789536 –768.7562887 –768.5650136 –768.5068132 –768.3334484 –768.287093 –768.2775959 –768.2540191 –768.2240512 –768.218499
0.00 43.67 46.44 58.07 93.26 118.33 127.49 132.71 149.31 176.75 1102.87 1112.1 1123.48 1124.7 1127.41 1128.11 1138.83 1139.01 1141.2 1147.84 1158.54 1159.64 1171.29 1173.1 1178.93 1179.13 1179.75 1184.96 1186.54 1191.26 1192.52 1192.78 1193.42 1194.23 1194.38
0.00 43.1688 47.0055 58.4933 93.2869 118.541 127.357 132.874 149.302 176.38 1102.72
0.00 43.16 47 58.49 93.28 118.54 127.35 132.87 149.3 176.38 1105.07 1112.77 1125.26
1125.28 1129.12 1140.53 1141.77 1142.14 1149.71 1152.43
1129.16 1129.72 1140.43 1140.63 1142.03 1149.62
1172.27 1174.1 1180.83 1181.07
1174.91 1180.15 1180.8
1193.35 1193.22 1194.96
1193.27 1193.12 1206.11
1195.7
1195.64
Reference [38] Theor.
Reference [38] Exp.
0.00 42.86 46.83 58.16 93.78 118.27 126.86 132.87 149.41 176.93 1105.84 1114.73 1126.46 1126.47 1130.22 1130.64 1141.79 1142.39 1143.35 1151.31 1160.33 1162.56 1173.74 1175.29 1181.23 1181.79 1181.94 1184.87 1189.09 1192.9 1194.45 1193.59 1206.61 1195.94 1196.42
0.00 43.13 46.99 58.51 93.31 118.44 127.26 132.84 149.42 176.42 1105.10 1112.80 1125.20 1125.18 1129.20 1129.69 1140.33 1140.96 1141.97 1149.65 1159.01 1161.19 1172.67 1174.46 1180.19 1180.88 1181.03 1183.43 1188.16 1192.22 1193.11 1193.56 1195.29 1195.17 1195.61
Reference [39] 0.00 43.219 47.046 58.538 93.308 118.599 127.412 132.937 149.341 176.383
Atoms 2015, 3
6
Table 2. Cont. Key
Configuration
J
Parity
Etotal
Present
NIST
Reference [26]
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
1s2 2p3d(3 D) 1s2 2p3p(1 D) 1s2 2p3d(3 D) 1s2 2p3d(3 F ) 1s2 2p3d(3 D) 1s2 2p3d(3 P ) 1s2 2p3d(3 P ) 1s2 2p3d(3 P ) 1s2 2p3p(1 S) 1s2 2p3d(1 F ) 1s2 2p3d(1 P ) 1s2 2s4s(3 S) 1s2 2s4s(1 S) 1s2 2s4p(3 P ) 1s2 2s4p(3 P ) 1s2 2s4p(3 P ) 1s2 2s4p(1 P ) 1s2 2s4d(3 D) 1s2 2s4d(3 D) 1s2 2s4d(3 D) 1s2 2s4d(1 D) 1s2 2s4f (3 F ) 1s2 2s4f (3 F ) 1s2 2s4f (1 F ) 1s2 2s4f (3 F ) 1s2 2p4s(3 P ) 1s2 2p4s(3 P ) 1s2 2p4p(3 D) 1s2 2p4p(3 D) 1s2 2p4p(3 P ) 1s2 2p4d(3 F ) 1s2 2p4d(3 P ) 1s2 2p4d(3 F ) 1s2 2p4d(3 D) 1s2 2p4f (3 G) 1s2 2p4s(3 P ) 1s2 2p4f (3 F ) 1s2 2p4f (3 G) 1s2 2p4f (3 D) 1s2 2p4s(1 P ) 1s2 2p4p(1 P ) 1s2 2p4p(3 P ) 1s2 2p4p(3 D) 1s2 2p4p(3 S) 1s2 2p4p(1 D) 1s2 2p4d(1 D) 1s2 2p4p(1 S) 1s2 2p4d(3 D) 1s2 2p4d(3 P ) 1s2 2p4d(3 P ) 1s2 2p4d(3 F ) 1s2 2p4f (3 F ) 1s2 2p4d(3 P ) 1s2 2p4f (3 F )
1 2 2 4 3 1 2 0 0 3 1 1 0 1 0 2 1 2 1 3 2 2 3 3 4 1 0 1 2 1 2 2 3 1 4 2 2 3 3 1 1 2 3 1 2 2 0 3 1 2 4 4 0 2
– + – – – – – – + – – + + – – – – + + + + – – – – – – + + + – – – – + – + + + – + + + + + – + – – – – + – +
–768.1717335 –767.9295858 –767.8457058 –767.8033102 –767.7047396 –767.6426327 –767.641839 –767.5905258 –767.5265103 –767.3347315 –767.3208321 –757.6906561 –757.5602444 –757.3878963 –757.370068 –757.3205243 –757.3072762 –757.1611615 –757.1547887 –757.1038753 –757.051647 –757.029191 –757.0029963 –756.9746835 –756.9456257 –755.8087174 –755.8024205 –755.6064384 –755.482694 –755.4767887 –755.3757501 –755.3004331 –755.2586403 –755.2580034 –755.241964 –755.229847 –755.2296258 –755.2292762 –755.2093178 –755.2038121 –754.9800163 –754.9598815 –754.9258014 –754.925425 –754.8394761 –754.7748959 –754.7230568 –754.7134465 –754.7071109 –754.7010038 –754.6940971 –754.6817078 –754.6731993 –754.6511138
1195.66 1202.25 1204.53 1205.68 1208.36 1210.05 1210.08 1211.47 1213.21 1218.43 1218.81 1480.86 1484.4 1489.09 1489.58 1490.93 1491.29 1495.26 1495.44 1496.82 1498.24 1498.85 1499.57 1500.34 1501.13 1532.06 1532.24 1537.57 1540.94 1541.1 1543.85 1545.9 1547.03 1547.05 1547.49 1547.82 1547.82 1547.83 1548.37 1548.52 1554.61 1555.16 1556.09 1556.1 1558.44 1560.2 1561.61 1561.87 1562.04 1562.21 1562.39 1562.73 1562.96 1563.56
1194.84 1203.76 1206.12 1208.72
1198.03 1203.61 1194.87 1205.09 1209.19
1209.22
1211.63
1218.8 1218.52
1218.61
1485.45 1490.79
1485.4 1491.06
1493.27 1497.11 1496.86 1497.85 1499.96
1493.23 1497.07 1496.83 1497.79 1499.93
1542.74
1547.82 1548.31
1547.36 1547.7 1548.22
1557.2
1557.12
1561.83
1557.12 1561.79
1562.57 1564.06 1563.94
1562.54 1564.02 1563.9
Reference [38] Theor.
Reference [38] Exp.
1198.49 1204.21 1196.01 1205.33 1209.8 1212.27 1212.18 1212.55 1214.11 1220.05 1221.25 1484.44 1487.3 1492.46 1492.24 1493.93 1494.55 1498.77 1498.57 1499.15 1501.61 1502.11 1502.21 1503.03 1502.39 1535.31 1534.72 1540.47 1543.87 1543.64 1546.68 1548.6 1549.2 1549.88 1551.14 1550.3 1550.98 1550.57 1551.08 1551.12 1556.65 1560.57 1557.48 1558.15 1557.52 1562.34 1564.05 1563.33 1564.28 1564.28 1562.34 1565.57 1564.42
1197.51 1203.62 1204.91 1206.13 1209.32 1211.77 1211.80 1212.07 1213.37 1218.89 1220.56 1482.28 1485.43 1490.46 1490.73 1492.42 1493.23 1497.02 1497.25 1497.68 1499.97 1500.45 1500.55 1500.75 1501.37 1532.53 1533.18 1538.88 1542.36 1542.64 1542.87 1545.15 1547.27 1547.79 1548.74 1548.90 1549.08 1549.50 1549.62 1549.62 1549.66 1555.67 1556.78 1556.62 1557.12 1559.90 1561.31 1561.67 1562.55 1563.46 1563.70 1563.66 1563.79
Reference [39]
Atoms 2015, 3
7
Table 2. Cont. Key
Configuration
J
Parity
Etotal
Present
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
1s2 2p4f (1 G) 1s2 2p4f (1 F ) 1s2 2p4f (3 D) 1s2 2p4f (3 D) 1s2 2p4f (1 D) 1s2 2p4d(1 P ) 1s2 2p4d(1 F ) 1s2 2s5s(3 S) 1s2 2s5s(1 S) 1s2 2s5p(3 P ) 1s2 2s5p(3 P ) 1s2 2s5p(3 P ) 1s2 2s5p(1 P ) 1s2 2s5d(3 D) 1s2 2s5d(3 D) 1s2 2s5d(1 D) 1s2 2s5d(3 D) 1s2 2s5d(3 D) 1s2 2s5g(3 G) 1s2 2s5g(1 G) 1s2 2s5f (3 F ) 1s2 2s5f (3 F ) 1s2 2s5f (1 F ) 1s2 2s5g(3 G) 1s2 2s5g(3 G) 1s2 2s5f (3 F ) 1s2 2p5s(3 P ) 1s2 2p5s(3 P ) 1s2 2p5p(3 D) 1s2 2p5p(3 P ) 1s2 2p5p(3 D) 1s2 2p5p(3 P ) 1s2 2p5d(3 F ) 1s2 2p5g(3 H) 1s2 2p5d(3 P ) 1s2 2p5d(3 D) 1s2 2p5f (3 G) 1s2 2p5f (3 F ) 1s2 2p5f (3 G) 1s2 2p5d(3 F ) 1s2 2p5f (3 D) 1s2 2p5g(3 G) 1s2 2p5g(3 H) 1s2 2p5g(3 F ) 1s2 2p5s(3 P ) 1s2 2p5s(1 P ) 1s2 2p5p(3 D) 1s2 2p5p(1 P ) 1s2 2p5p(3 S) 1s2 2p5p(3 D) 1s2 2p5p(1 D) 1s2 2p5d(1 D) 1s2 2p5p(1 S)
4 3 3 1 2 1 3 1 0 1 0 2 1 2 1 2 1 3 4 4 2 3 3 3 5 4 1 0 1 0 2 1 2 5 2 1 4 2 3 3 3 3 4 4 2 1 2 1 1 3 2 2 0
+ + + + + – – + + – – – – + + + + + + + – – – + + – – – + + + + – – – – + + + – + – – – – – + + + + + – +
–754.6473417 –754.6449993 –754.6115814 –754.603951 –754.5999827 –754.5893505 –754.5841823 –751.392181 –751.3292255 –751.246659 –751.221321 –751.2099355 –751.2011308 –751.1374152 –751.126486 –751.0866158 –751.126486 –751.0860762 –751.0857587 –751.0826556 –751.0680262 –751.041488 –751.0238241 –751.0218288 –751.0212189 –750.9786648 –749.564771 –749.5389946 –749.4542569 –749.4006235 –749.3994227 –749.3905489 –749.3402395 –749.3026955 –749.2983472 –749.2867415 –749.2824452 –749.2733538 –749.2659755 –749.2653948 –749.2555687 –749.2352951 –749.1684219 –749.1647046 –748.9871642 –748.9732118 –748.8524328 –748.8511734 –748.8228875 –748.8108735 –748.7936606 –748.7499592 –748.7228871
1563.67 1563.73 1564.64 1564.85 1564.95 1565.24 1565.38 1652.24 1653.95 1656.2 1656.89 1657.2 1657.44 1659.17 1659.47 1660.56 1659.47 1660.57 1660.58 1660.66 1661.06 1661.78 1662.26 1662.32 1662.34 1663.49 1701.97 1702.67 1704.97 1706.43 1706.47 1706.71 1708.08 1709.1 1709.22 1709.53 1709.65 1709.9 1710.1 1710.11 1710.38 1710.93 1712.75 1712.85 1717.68 1718.06 1721.35 1721.38 1722.15 1722.48 1722.95 1724.14 1724.88
NIST
1566.04
Reference [26]
1563.96 1565.88 1655.94 1659.28
1659.28 1661 1657.54 1666.1 1661.88
1711.48
1723.88 1726.11
1659.28 1661.34 1666.05 1659.43 1661.77
Reference [38] Theor.
Reference [38] Exp.
1566.63 1565.24
1565.69 1564.31
1566.91 1566.07 1567.36 1566.78 1653.77 1655.09 1657.83 1657.73 1658.66 1659.03 1661.04 1660.94 1662.22 1660.94 1661.23 1662.95 1663.04 1662.78 1662.81 1663.23 1662.95 1663.02 1662.9 1703.76 1703.54 1706.64 1708.4 1708.39 1708.26 1709.85 1712.02 1710.88 1711.22 1712.13 1712.07 1711.84 1711.07 1712.11 1711.97 1711.93 1712.05 1719.42 1719.89
1566.04 1565.16 1566.94 1566.04
1723.03 1723.76 1723.55 1723.32 1725.94 1726.27
Reference [39]
Atoms 2015, 3
8
Table 2. Cont. Key
Configuration
J
Parity
Etotal
Present
143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182
1s2 2p5d(3 P ) 1s2 2p5d(3 P ) 1s2 2p5f (3 F ) 1s2 2p5d(3 D) 1s2 2p5f (3 F ) 1s2 2p5f (1 G) 1s2 2p5d(3 P ) 1s2 2p5f (1 F ) 1s2 2p5f (1 D) 1s2 2p5d(1 P ) 1s2 2p5g(3 F ) 1s2 2p5f (3 D) 1s2 2p5d(3 F ) 1s2 2p5f (3 D) 1s2 2p5g(3 G) 1s2 2p5g(3 G) 1s2 2p5d(1 F ) 1s2 2p5g(1 H) 1s2 2p5g(3 F ) 1s2 2p5f (3 G) 1s2 2p5g(3 G) 1s2 2p5g(3 F ) 1s2 2p4p(3 P ) 1s2 3s2 (1 S) 1s2 3s3p(3 P ) 1s2 3s3p(3 P ) 1s2 3s3p(3 P ) 1s2 3s3p(1 P ) 1s2 3s3d(1 D) 1s2 3s3d(3 D) 1s2 3s3d(3 D) 1s2 3s3d(3 D) 1s2 3p2 (3 P ) 1s2 3p2 (3 P ) 1s2 3p2 (3 P ) 1s2 3p3d(3 F ) 1s2 3p3d(3 F ) 1s2 3p3d(1 D) 1s2 3p2 (1 D) 1s2 3p2 (1 S)
1 2 4 3 2 4 0 3 2 1 2 1 4 3 3 5 3 5 3 5 4 4 0 0 0 1 2 1 2 1 2 3 0 1 2 2 3 2 2 0
– – + – + + – + + – – + – + – – – – – + – – + + – – – – + + + + + + + – – – + +
–748.7220533 –748.7152748 –748.7106954 –748.7069321 –748.6949316 –748.6906284 –748.6802854 –748.6721365 –748.6702054 –748.6683712 –748.6669093 –748.6651871 –748.658763 –748.6564126 –748.6544227 –748.6511383 –748.6435973 –748.6425123 –748.63994 –748.6322597 –748.599593 –748.5922492 –746.0926537 –729.7643403 –729.3728951 –729.3646407 –729.2408485 –728.8209355 –728.7962086 –728.6774571 –728.667205 –728.604182 –728.5677089 –728.5185534 –728.4270966 –728.2811758 –728.1545448 –728.1238826 –728.0448765 –728.0435435
1724.9 1725.08 1725.21 1725.31 1725.64 1725.75 1726.03 1726.26 1726.31 1726.36 1726.4 1726.45 1726.62 1726.68 1726.74 1726.83 1727.03 1727.06 1727.13 1727.34 1728.23 1728.43 1796.45 2240.75 2251.4 2251.63 2255 2266.42 2267.1 2270.33 2270.61 2272.32 2273.31 2274.65 2277.14 2281.11 2284.56 2285.39 2287.54 2287.58
NIST
Reference [26]
Reference [38] Theor.
Reference [38] Exp.
Reference [39]
1726.75
1726.98
1727.69 1726.34 1728.16 1726.82 1727.54 1727.89 1728.05 1728.1 1728.25 1725.99
1728.96
1727.75 1727.74 1727.95 1728.18 1727.98 1727.87
One might wonder whether quadruple excitations are really necessary. Two examples illustrate the improvement of the results when quadruple excitations are included. The first example is level 14, 1s2 2s3p(3 P0 ), for which the present calculation gives an energy of 1124.70 eV which agrees to within less than 0.2% with the value 1126.47 eV recorded in the CHIANTI database [38]. If only triple excitations are included in the calculations the level energy drops to 1109.07 eV with an increase of the deviation from the CHIANTI energy by as much as 1.5%. The second example, that we want to mention, is level 21, 1s2 2p3s(3 P0 ), which has a calculated energy of 1158.54 eV. This corresponds reasonably well to the NIST energy [34] 1152.40 eV with a relative difference of less than 0.55%. Again, when only triple excitations are considered, the level energy drops drastically, in this case to 1143.12 eV producing a difference in the resulting level energy of 15.42 eV and a relative deviation from the NIST energy
Atoms 2015, 3
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of about 0.9%. Obviously, the agreement of the calculations with the CHIANTI and NIST databases becomes considerably better with the use of 4-electron excitations. Dirac-Fock wave functions with a minimum number of radial functions are not sufficient to represent the occupied orbitals. Extra configurations have to be added to adequately represent electron correlations (i.e., mixing coefficients). These extra configurations are represented by CSFs and must have the same angular momentum and parity as the occupied orbital [40]. For instance, the level 1s2 2s2p(3 P0− ) is represented by 0.996 of 1s2 2s2p(3 P0− ) and 0.0660 of 1s2 2s3p(3 P0− ). The mixing coefficients for the wave functions of some calculated levels are shown in Table 3. The most important contributions to the total wave function of a given level are those from the same configuration. For example, the configuration-mixed wave function for the 1s2 2p3p(3 P0 ) level is represented as |1s2 2p3p(3 P0 )i = 0.9501|1s2 2p3p(3 P0 )i + 0.3|1s2 2p3p(1 S0 )i − 0.0697|1s2 2s3s(1 S0 )i where 0.9501, 0.3, and −0.0697 are the configuration mixing coefficients. Coefficients less than 0.05 were calculated but are not explicitly given. Expansion coefficients for several levels by Bhatia and Mason [10] are listed in Table 3 for comparison. Clearly, the present and the previous [10] results are very close to one another in the description of the configuration-interaction wave functions. Table 3. The configuration mixing coefficients (> 0.05) for some levels in Fe22+ . The number in the bra-kets refers to the level number (the key in Table 2). JP Configuration
Present Work
Reference [10]
J = 0+ 1s2 2s2 (1 S) 1s2 2p2 (3 P ) 1s2 2p2 (1 S) 1s2 2s3s(1 S) 1s2 2p3p(3 P )
0.9772 |1i + 0.2058 |10i 0.9663 |6i + 0.2431 |10i − 0.075 |1i 0.9465 |10i − 0.2533 |6i − 0.19241 |1i 0.9814 |12i + 0.1688 |44i − 0.0708|48i 0.9501 |28i + 0.3 |44i − 0.0697 |12i
0.98 |1i + 0.21 |10i 0.97 |6i + 0.24 |10i − 0.07 |1i 0.95 |10i − 0.25 |6i − 0.19 |1i 0.99 |12i + 0.17 |44i
J = 1+ 1s2 2p2 (3 P ) 1s2 2s3s(3 S) 1s2 2s3d(3 D) 1s2 2p3p(3 D) 1s2 2p3p(1 P )
0.9992 |7i 0.9913 |11i + 0.1301 |34i 0.9896 |17i + 0.1399 |23i 0.8202 |36i + 0.5328 |27i − 0.1612 |34i + 0.1265 |17i 0.5302 |27i + 0.5204 |34i + 0.4903 |30i + 0.4449 |23i
1.00 |7i 0.99 |11i + 0.13 |34i 0.99 |17i + 0.14 |23i
J = 2+ 1s2 2p2 (3 P ) 1s2 2p2 (1 D) 1s2 2s3d(3 D) 1s2 2s3d(1 D) 1s2 2p3p(3 D)
0.8640 |8i − 0.5014 |9i 0.8637 |9i + 0.5016 |8i 0.9903 |18i − 0.1169 |26i 0.9895 |20i − 0.1180 |37i 0.9169 |26i − 0.2744 |37i + 0.2595 |35i + 0.1143 |18i
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Table 3. Cont. JP Configuration
Present Work
Reference [10]
J = 3+ 1s2 2s3d(3 D) 1s2 2p3p(3 D) 1s2 2s4d(3 D) 1s2 2p4f (3 G)
0.9939 |19i − 0.0943 |32i 0.9948 |32i + 0.0939 |19i 0.9976 |55i 0.8423 |73i + 0.4515 |91i + 0.2737 |89i
J = 4+ 1s2 2p4f (3 G) 1s2 2p4f (3 F ) 1s2 2p4f (1 G) 1s2 2s5g(3 G)
0.6431 |70i − 0.5492 |90i + 0.5300 |87i 0.8133 |87i − 0.5656 |70i + 0.1220 |90i 0.8241 |90i + 0.5122 |70i + 0.2327 |87i 0.8292 |108i + 0.5561 |109i
J = 5+ 1s2 2s5g(3 G) 1s2 2p5f (3 G)
0.9982 |114i 0.9980 |162i
J = 0− 1s2 2s2p(3 P ) 1s2 2s3p(3 P ) 1s2 2p3s(3 P ) 1s2 2p3d(3 P )
0.9960 |2i + 0.0660 |14i 0.9908 |14i − 0.0821 |43i 0.9927 |21i + 0.0714 |43i 0.9918 |43i + 0.0860 |14i
1.00|2i
J = 1− 1s2 2s2p(3 P ) 1s2 2s2p(1 P ) 1s2 2s3p(3 P ) 1s2 2s3p(1 P )
0.9840 |3i + 0.1564 |5i 0.9833 |5i − 0.1568 |3i 0.7772 |13i + 0.5935 |15i − 0.1644 |25i 0.7411 |15i − 0.6127 |13i − 0.238 |25i
0.99 |3i + 0.16 |5i 0.99 |5i − 0.16 |3i 0.80 |13i + 0.58 |15i − 0.16 |25i 0.76 |15i − 0.59 |13i − 0.24 |25i
J = 2− 1s2 2s2p(3 P ) 1s2 2s3p(3 P ) 1s2 2p3s(3 P ) 1s2 2p3d(3 F )
0.9965 |4i 0.9890 |16i 0.9843 |24i 0.8727 |29i + 0.4391 |33i + 0.1375 |38i
J = 3− 1s2 2p3d(3 F ) 1s2 2p3d(3 D) 1s2 2p3d(1 F ) 1s2 2s4f (3 F )
0.8829 |31i + 0.3724 |40i − 0.2776 |45i 0.8737 |40i − 0.4342 |31i − 0.2094 |45i 0.9344 |45i + 0.3064 |40i + 0.1649 |31i 0.9807 |58i + 0.1863 |59i
J = 4− 1s2 23d(3 F ) 1s2 2s4f (3 F ) 1s2 2p4d(3 F )
0.9970 |39i 0.9983 |60i 0.9973 |86i
J = 5− 1s2 2p5g(3 G) 1s2 3d4f (3 H)
0.8469 |158i − 0.4368 |2p5g(3 H)i + 0.2964 |160i 0.9974 |268i
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A comparison between the calculated wavelengths and other published experimental and theoretical values [1,2,8,24,26,28,29,34,41] is shown in Table 4. The accuracy of calculated wavelengths (in Å) relative to measurements [24,26,34] can be assessed from Table 4, where the agreement is within
